Infinite-size Matrix Research Articles

Dislocation Dipole in a Strained Nanostructured Layer Buried in an Infinite-Size Matrix

Abstract The equilibrium positions of a dipole of edge dislocations embedded in a nanostructure composed of a strained two-dimensional layer buried in an infinite-size matrix with a rectangular nanowire located at the upper layer–matrix interface have been determined from a Peach–Koëhler force calculation. The location of the unstable and stable equilibrium positions of the dislocations of the dipole gliding in the horizontal plane has been specified in the nanostructure versus the misfit strain and nanowire aspect ratio. The critical misfit below which the dipole has no equilibrium position has been finally determined versus the size of the nanowire.

Effect of stiffness contrast on the dislocation equilibrium positions in a multi-layered structure

The equilibrium positions of two edge dislocations lying in the same gliding plane have been theoretically determined in a structure composed of a thin layer embedded in an infinite-size matrix of different elastic constants, when the two dislocations are symmetrically distributed with respect to the axis of symmetry of the problem parallel to the interfaces. When the Burgers vectors of the two dislocations are equal and the matrix is stiffer than the layer, the equilibrium positions have been found to be unstable in the central layer and stable in the matrix. No equilibrium position is found when the matrix is softer than the layer. When the Burgers vectors are of opposite sign (same norm and direction) and the layer is stiffer than the matrix, the equilibrium positions are stable in the central layer and unstable in the matrix, no equilibrium position being found when the layer is softer than the matrix.

Entanglement entropy of two-species hard-core bosons in one dimension

We study the von Neumann entropy of a model for two-species hard-core bosons in one dimension. In this model, the same-species bosons satisfy hard-core conditions, while the different-species bosons are allowed to occupy the same site with a local interaction U . At half-filling, by Jordan-Wigner transformation, the model is exactly mapped to a fermionic Hubbard model. The phase transition from superfluid U = 0 to Mott insulator U > 0 can be explained by simple one-band theory at half-filling. We measure the von Neumann entanglement entropy of the ground states for the half-filled case and away from half-filling to understand quantum phase transitions. To achieve this goal, we use a time-evolution-block decimation method with infinite-size matrix product state and also use a density matrix renormalization group with matrix product operators with large bond dimensions up to 300. We found strong evidence that the local minimum point of the von Neumann entanglement entropy is the quantum critical point for finite-bond-dimension matrix product states.

Fracture Formation in Axisymmetrical Layered Materials

A stress-based criterion for the formation of a periodic distribution of cracks in an infinite-length cylindrical inclusion of radius R embedded in an infinite-size matrix has been established when the inclusion undergoes intrinsic strain. In agreement with previous studies, it is found that the distance separating two consecutive circular cracks of the same radius than that of inclusion does not depend on stress nor elastic coefficients of the material. This critical distance has been found to be of the order of 1.67 R.

Generation of a dipole of misfit dislocations in an axisymmetrical precipitate embedded in a semi-infinite matrix

The strain relaxation of an axisymmetrical precipitate embedded in a semi-infinite matrix with a planar free-surface has been theoretically investigated through the generation in the structure of a dislocation dipole. An energy variation calculation has been performed and the formation of the dipole composed of two misfit edge dislocations has been characterised in the interface. It is found that, contrary to the case where the inclusion is embedded in an infinite-size matrix, there exists near the free-surface a preferential plane selected for the dipole formation in the interface which corresponds to the plane perpendicular to the free surface and containing the inclusion centre. A stability diagram of the structure with respect to the dipole formation in the interface has been provided as a function of the misfit strain and the distance of the inclusion from the free-surface, for different values of the inclusion radius.

Global quantum discord in matrix product states and the application

Global quantum discord (GQD) is a measure of quantum correlation for multi-site quantum states. In this paper, we propose an efficient procedure to calculate global quantum discord (Gn) for consecutive n-site subsystems in infinite-size matrix product states (MPSs). We apply the method to study the scaling behavior of Gn in several one-dimensional infinite quantum spin chains, i.e., (1) a three-site interaction model whose ground state can be exactly expressed as MPSs, and (2) a spin-12 XXZ chain whose ground state is approximately expressed as MPSs with the help of infinite time-evolving block decimation (iTEBD) algorithm. In both models, as the increase of n, Gn shows an approximately linear growth. Unambiguous clue for the convergence of the incremental ΔGn=Gn−Gn−1 is observed when n is large enough. Moreover, in non-critical (gapped) regions ΔGn converges very fast, while in critical (gapless) regions it converges relatively slow. The behaviors are explained in a model-independent physical picture with finite-range correlations. Based on these results, we propose to use “discord per site” (Gnn) to describe the global correlations in infinite-size spin chains. Moreover, we find that the global discord shows a size-independent maximum at the infinite-order quantum phase transition point of the XXZ model.

Molecular Hubbard Hamiltonian: Field regimes and molecular species

The molecular Hubbard Hamiltonian (MHH) naturally arises for ultracold ground-state polar alkali-metal dimer molecules in optical lattices. We show that, unlike ultracold atoms, different molecules display different many-body phases due to intrinsic variances in molecular structure even when the molecular symmetry is the same. We also demonstrate a wide variety of experimental controls on ${}^{1}\ensuremath{\Sigma}$ molecules via external fields, including applied static electric and magnetic fields, an ac microwave field, and the polarization and strength of optical lattice beams. We provide explicit numerical calculations of the parameters of the MHH, including tunneling and direct and exchange dipole-dipole interaction energies, for the molecules ${}^{6}$Li${}^{133}$Cs, ${}^{23}$Na${}^{40}$K, ${}^{87}$Rb${}^{133}$Cs, ${}^{40}$K${}^{87}$Rb, and ${}^{6}$Li${}^{23}$Na in weak and strong applied electric fields. As case studies of many-body physics, we use infinite-size matrix product state methods to explore the quantum phase transitions from the superfluid phase to half-filled and third-filled crystalline phases in one dimension.

On the Surface Stability of a Spherical Void Embedded in a Stressed Matrix

The linear stability analysis of the shape of a spherical cavity embedded in an infinite-size matrix under stress has been performed when infinitesimal perturbation from sphericity of the rod is assumed to appear by surface diffusion. Developing the perturbation on a basis of complete spherical harmonics, the growth rate of each harmonic Ylm(θ,φ) has been determined and the conditions for the development of the different fluctuations have been discussed as a function of the applied stress and the order l of the perturbation.

Morphological instabilities of stressed axi-symmetrical structures embedded in a matrix: volume diffusion approach

The morphological evolution of an epitaxially stressed precipitate embedded in an infinite size matrix has been studied when vacancies are assumed to diffuse through the matrix/precipitate interface. The linear stability of the surface has been first investigated in front of longitudinal fluctuations and the influence of epitaxial stress has been characterized onto the development of the roughness already observed when the structure is traction-free. This linear analysis has been then extended to the case of radial perturbations of the interface. It has been observed that the interface, assumed to be stable when the cylinder is traction-free, becomes unstable when the stress is considered.